Quasicrystal

The first representations of perfect quasicrystalline patterns can be found in several early Islamic works of art and architecture such as the Gunbad-i-Kabud tomb tower, the Darb-e Imam shrine and the Al-Attarine Madrasa. On July 16, 1945, in Alamogordo, New Mexico, the Trinity nuclear bomb test produced icosahedral quasicrystals. They went unnoticed at the time of the test but were later identified in samples of red trinitite, a glass-like substance formed from fused sand and copper transmission lines. Identified in 2021, they are the oldest known anthropogenic quasicrystals. In 1961, Hao Wang asked whether determining if a set of tiles admits a tiling of the plane is an algorithmically unsolvable problem or not. He conjectured that it is solvable, relying on the hypothesis that every set of tiles that can tile the plane can do it periodically (hence, it would suffice to try to tile bigger and bigger patterns until obtaining one that tiles periodically). Nevertheless, two years later, his student Robert Berger constructed a set of some 20,000 square tiles (now called Wang tiles) that can tile the plane but not in a periodic fashion. As further aperiodic sets of tiles were discovered, sets with fewer and fewer shapes were found. In 1974 Roger Penrose discovered a set of just two tiles, now referred to as Penrose tiles, that produced only non-periodic tilings of the plane. These tilings displayed instances of fivefold symmetry. One year later Alan Mackay showed theoretically that the diffraction pattern from the Penrose tiling had a two-dimensional Fourier transform consisting of sharp 'delta' peaks arranged in a fivefold symmetric pattern. Shechtman related his observation to Ilan Blech, who responded that such diffractions had been seen before. Around that time, Shechtman also related his finding to John W. Cahn of the NIST, who did not offer any explanation and challenged him to solve the observation. Shechtman quoted Cahn as saying: "Danny, this material is telling us something, and I challenge you to find out what it is". The observation of the ten-fold diffraction pattern lay unexplained for two years until the spring of 1984, when Blech asked Shechtman to show him his results again. A quick study of Shechtman's results showed that the common explanation for a ten-fold symmetrical diffraction pattern, a type of crystal twinning, was ruled out by his experiments. Therefore, Blech looked for a new structure containing cells connected to each other by defined angles and distances but without translational periodicity. He decided to use a computer simulation to calculate the diffraction intensity from a cluster of such a material, which he termed as "multiple polyhedral", and found a ten-fold structure similar to what was observed. The multiple polyhedral structure was termed later by many researchers as icosahedral glass. Shechtman accepted Blech's discovery of a new type of material and chose to publish his observation in a paper entitled "The Microstructure of Rapidly Solidified Al6Mn", which was written around June 1984 and published in a 1985 edition of Metallurgical Transactions A. Meanwhile, on seeing the draft of the paper, John Cahn suggested that Shechtman's experimental results merit a fast publication in a more appropriate scientific journal. Shechtman agreed and, in hindsight, called this fast publication "a winning move". This paper, published in the Physical Review Letters, The term "quasicrystal" was first used in print by Paul Steinhardt and Dov Levine In 1992, the International Union of Crystallography altered its definition of a crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic. fragment. The corresponding diffraction patterns reveal a ten-fold symmetry. A further study of Khatyrka meteorites revealed micron-sized grains of another natural quasicrystal, which has a ten-fold symmetry and a chemical formula of Al71Ni24Fe5. This quasicrystal is stable in a narrow temperature range, from 1120 to 1200 K at ambient pressure, which suggests that natural quasicrystals are formed by rapid quenching of a meteorite heated during an impact-induced shock. In 2014, Post of Israel issued a stamp dedicated to quasicrystals and the 2011 Nobel Prize. While the first quasicrystals discovered were made out of intermetallic components, later on quasicrystals were also discovered in soft-matter and molecular systems. Soft quasicrystal structures have been found in supramolecular dendrimer liquids and ABC Star Polymers in 2004 and 2007. In 2009, it was found that thin-film quasicrystals can be formed by self-assembly of uniformly shaped, nano-sized molecular units at an air-liquid interface. It was demonstrated that these units can be both inorganic and organic. Additionally in the 2010s, two-dimensional molecular quasicrystals were discovered, driven by intermolecular interactions and interface-interactions. In 2018, chemists from Brown University announced the successful creation of a self-constructing lattice structure based on a strangely shaped quantum dot. While single-component quasicrystal lattices have been previously predicted mathematically and in computer simulations, they had not been demonstrated prior to this. ==Mathematics==

as an orthographic projection into 2D using Petrie polygon basis vectors overlaid on the diffractogram from an icosahedral Ho–Mg–Zn quasicrystal projected into the rhombic triacontahedron using the golden ratio in the basis vectors. This is used to understand the aperiodic icosahedral structure of quasicrystals. There are several ways to mathematically define quasicrystalline patterns. One definition, the "cut and project" construction, is based on the work of Harald Bohr (mathematician brother of Niels Bohr). The concept of an almost periodic function (also called a quasiperiodic function) was studied by Bohr, including work of Bohl and Escanglon. He introduced the notion of a superspace. Bohr showed that quasiperiodic functions arise as restrictions of high-dimensional periodic functions to an irrational slice (an intersection with one or more hyperplanes), and discussed their Fourier point spectrum. These functions are not exactly periodic, but they are arbitrarily close in some sense, as well as being a projection of an exactly periodic function. In order that the quasicrystal itself be aperiodic, this slice must avoid any lattice plane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed as two-dimensional slices of five-dimensional hypercubic structures; similarly, icosahedral quasicrystals in three dimensions are projected from a six-dimensional hypercubic lattice, as first described by Peter Kramer and Roberto Neri in 1984. Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors, which are the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice. Using mathematics for construction and analysis of quasicrystal structures is a difficult task. Computer modeling, based on the existing theories of quasicrystals, however, greatly facilitated this task. Advanced programs have been developed It is suggested that the electronic system of some quasicrystals is located at a quantum critical point without tuning, while quasicrystals exhibit the typical scaling behaviour of their thermodynamic properties and belong to the well-known family of heavy fermion metals. ==Materials science==

formed as a pentagonal dodecahedron, the dual of the icosahedron. Unlike the similar pyritohedron shape of some cubic-system crystals such as pyrite, the quasicrystal has faces that are true regular pentagons Since the original discovery by Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Quasicrystals are found most often in aluminium alloys (Al–Li–Cu, Al–Mn–Si, Al–Ni–Co, Al–Pd–Mn, Al–Cu–Fe, Al–Cu–V, etc.), but numerous other compositions are also known (Cd–Yb, Ti–Zr–Ni, Zn–Mg–Ho, Zn–Mg–Sc, In–Ag–Yb, Pd–U–Si, etc.). The molecular compound ferrocenecarboxylic acid is also known to form a quasicrystal in two dimensions. Two types of quasicrystals are known. Quasicrystals fall into three groups of different thermal stability: • Stable quasicrystals grown by slow cooling or casting with subsequent annealing, • Metastable quasicrystals prepared by melt spinning, and • Metastable quasicrystals formed by the crystallization of the amorphous phase. Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by X-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions. The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals. A nanoscale icosahedral phase was formed in Zr-, Cu- and Hf-based bulk metallic glasses alloyed with noble metals. Most quasicrystals have ceramic-like properties including high thermal and electrical resistance, hardness and brittleness, resistance to corrosion, and non-stick properties. While the softcore Rydberg dressing interaction has forms triangular droplet-crystals, adding a Gaussian peak to the plateau type interaction would form multiple roton unstable points in the Bogoliubov spectrum. Therefore, the excitation around the roton instabilities would grow exponentially and form multiple allowed lattice constants leading to quasi-ordered periodic droplet crystals. as a coating for frying pans. Food did not stick to it as much as to stainless steel making the pan moderately non-stick and easy to clean; heat transfer and durability were better than PTFE non-stick cookware and the pan was free from perfluorooctanoic acid (PFOA); the surface was very hard, claimed to be ten times harder than stainless steel, and not harmed by metal utensils or cleaning in a dishwasher; and the pan could withstand temperatures of without harm. However, after an initial introduction the pans were a chrome steel, probably because of the difficulty of controlling thin films of the quasicrystal. The Nobel citation said that quasicrystals, while brittle, could reinforce steel "like armor". When Shechtman was asked about potential applications of quasicrystals he said that a precipitation-hardened stainless steel is produced that is strengthened by small quasicrystalline particles. It does not corrode and is extremely strong, suitable for razor blades and surgery instruments. The small quasicrystalline particles impede the motion of dislocation in the material. Other potential applications include selective solar absorbers for power conversion, broad-wavelength reflectors, and bone repair and prostheses applications where biocompatibility, low friction and corrosion resistance are required. Magnetron sputtering can be readily applied to other stable quasicrystalline alloys such as Al–Pd–Mn. ==Non-material science applications==

Applications in macroscopic engineering have been suggested, building quasi-crystal-like large scale engineering structures, which could have interesting physical properties. Also, aperiodic tiling lattice structures may be used instead of isogrid or honeycomb patterns. The mechanical properties of honeycombs, based on a wide range of aperiodic tilings, have been investigated through simulation and physical testing of samples produced by 3D printing. The quasicrystal inspired honeycombs offer a broader range of mechanical properties than is available from commonly used honeycombs and are generally isotropic in the plane. Of particular note are honeycombs based on the aperiodic monotile, discovered by David Smith et al. These honeycombs offer planar isotropic zero Poisson's ratio, a rare elastic property that removes orthogonal deformation in response to an applied force. And, it has been shown that composite materials constructed of a hard matrix and a network of soft material mimicking the pattern of the aperiodic monotile can offer improved strength and toughness compared to similar composites using other patterns. ==See also==